Abstract
The Couette problem is the simplest problem of steady shear flow of rarefied gas in a region bounded by solid surfaces. This problem has been examined in the linear formulation by many authors, using either the linearized Krook equation or the moment methods (see [1]). It has recently been solved by the Monte Carlo method [2].
The nonlinear problem of Couette flow with heat transfer for the Krook equation has been solved by reducing the problem to a system of integral equations [3] over a wide range of flat-plate velocities and temperature ratios and by the discrete-velocity method [4] for moderate plate velocities. In this article we solve the same problem for the generalized Krook equation [5] which approximates the Boltzmann equation for a pseudo-Maxwellian gas in accordance with the method suggested by the author [6, 7]. The generalized Krook equation was solved numerically by a modified discrete-velocity method which has been used by the author previously to solve the problem of shock wave structure [8].
The primary case examined is that of pseudo-Maxwellian molecules, in which the viscosity is proportional to the temperature. The computations were made for Prandtl numbers of 1 and 2/3 over a wide range of Mach and Knudsen numbers as well as flat-plate temperature ratios. As we would expect, the Prandtl number effect is greatest for small Knudsen numbers. The flow velocity profiles are not very sensitive to variation of the Prandtl number (at least for pseudo-Maxwellian molecules).
However, the most interesting result of the study is independent of the Prandtl number. Specifically, it was found that for any sufficiently high flat-plate velocities the friction stress, referred to the corresponding free molecular value, does not change monotonically with variation of the Knudsen number; instead, there is a peak. As far as the author is aware, this nonlinear effect has not been discussed previously in the literature (including articles [3, 4]).
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References
M. N. Kogan, Rarefied Gasdynamlcs, Kinetic Theory [in Russian], Nauka, Moscow, 1967.
S. L. Gorelov and M. N. Kogan, “Solution of linear problems of rarefied gasdynamics by the Monte Carlo method”, Izv. AN SSSR. MZhG [Fluid Dynamics], vol. 3, no. 6, 1968.
D. G. Anderson, “Numerical solutions of the Krook kinetic equation”, J. Fluid Mech., vol. 25, pt 2, pp. 271–287, 1966.
A. B. Huang and D. L. Hartley, “Nonlinear rarefied Couette flow with heat transfer”, Phys. Fluids, vol. 2, no. 6, pp. 1321–1326, 1968.
E. M. Shakhov, “An extension of the Krook relaxational kinetic equation”, Izv. AN SSSR. MZhG [Fluid Dynamics], vol. 3, no. 5, 1968.
E. M. Shakhov, “The approximate kinetic equations in rarefied gas theory”, Izv. AN SSSR. MZhG [Fluid Dynamics], vol. 3, no. 1, 1968.
E. M. Shakhov, “Method for approximating the Boltzmann kinetic equation”, collection: Numerical Methods in Rarefied-Gas Theory [in Russian], Tr. VTs AN SSSR, 1969.
E. M. Shakhov, “The structure of the compression shock in a monatomic gas”, Izv. AN SSSR. MZhG [Fluid Dynamics], vol. 4, no. 2, 1969.
D. R. Willis, “Heat transfer in a rarefied gas between parallel plates at large temperature ratios” collection: Rarefied Gasdynamics, Acad. Press, New York-London, Vol. 1, 1963.
C. Cercignani and G. Tironi, “Nonlinear heat transfer between two parallel plates according to a model with correct Prandtl number”, collection: Rarefied Gasdynamics, Vol. 1, Acad. Press, New York-London, 1967.
L. H. Holway, “New statistical models in kinetic theory: methods of construction”, Phys. Fluids, vol. 9, no. 9, pp. 1658–1673, 1966.
D. R. Willis, “Theoretical solutions of some nearly free molecular problems”, collection: Rarefied Gasdynamics, Pergamon Press, New York-London, 1960.
D. R. Willis, “Comparison of kinetic theory analysis of linearized Couette flow”, Phys. Fluids, vol. 5, no. 8, pp. 127–135, 1962.
C. Y. Liu and L. Lees, “Kinetic theory description of plane compressible Couette flow”, collection: Rarefied Gasdynamics, Acad. Press, 1961.
M. Abramowitz, Evaluation of the integral\(\int\limits_0^\infty {e^{ - u2 - x/u} } \)du, J. Math. and Phys., vol. 32, no. 2, 3, pp. 188–192, 1953.
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Shakhov, E.M. Couette problem for the generalized Krook equation stress-peak effect. Fluid Dyn 4, 9–13 (1969). https://doi.org/10.1007/BF01015946
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DOI: https://doi.org/10.1007/BF01015946